论文标题
多维Schrödinger操作员的频谱具有半线和cantor套装
Multidimensional Schrödinger Operators Whose Spectrum Features a Half-Line and a Cantor Set
论文作者
论文摘要
我们构建具有高能量差异的频谱的多维Schrödinger运算符，在低能量下，这无处不在。这给出了第一个示例，可以证实，在统一的经常性Schrödinger操作员类别中，这种广泛期望的拓扑结构可以证实半线和康托型结构的共存。我们的构造使用具有可分离电位的Schrödinger运算符，可分解为斐波那契序列产生的一维电势，并通过痕量图和Fricke-vogt不变性依赖于此类操作员的研究。为了证明频谱包含半线，我们证明了一个抽象的伯特贝尔菲尔德标准，用于可能具有独立关注的cantor集合。
We construct multidimensional Schrödinger operators with a spectrum that has no gaps at high energies and that is nowhere dense at low energies. This gives the first example for which this widely expected topological structure of the spectrum in the class of uniformly recurrent Schrödinger operators, namely the coexistence of a half-line and a Cantor-type structure, can be confirmed. Our construction uses Schrödinger operators with separable potentials that decompose into one-dimensional potentials generated by the Fibonacci sequence and relies on the study of such operators via the trace map and the Fricke-Vogt invariant. To show that the spectrum contains a half-line, we prove an abstract Bethe--Sommerfeld criterion for sums of Cantor sets which may be of independent interest.
